We blackjointly estimate VaR and ES of the resulting portfolios and analyze their out-of-sample performance using the last 368 returns of the sample. The backtesting results for the considered strategies are shown in Table REF , where we report the AL log-score of [1]}, together with the S\(_{FZN}\)  and S\(_{FZ0}\)  loss functions in (REF ) and (REF ) for the joint evaluation of the pair (VaR, ES). The results clearly make our approach stand out compared to the other strategies. Indeed, blackthe strategies based on the multivariate Normal and t- distributions, and their skewed counterparts, produce highly volatile VaR forecasts and suffer from larger average losses over the out-of-sample period. On the other hand, the SMV and Moment-SMV models deliver better performance gains over the MV portfolios, with the SMV being preferred at the three VaR levels, especially at the most extreme case of \(\tau = 0.01\) . Such gain may be traced back to the higher efficiency in the estimation procedure based on the ML approach proposed in Section REF . It is worth noticing that these conclusions remain still valid even when we use the \(S_{FZ0}\)  and the \(S_{FZN}\)  scoring rules, which do not directly depend on the AL likelihood function.
